PHY183-Lecture5 - Physics for Scientists &...

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Unformatted text preview: Physics for Scientists & Engineers 1 Spring Semester 2006 Lecture 5 January 20, 2006 Physics for Scientists&Engineers 1 1 Announcements Course Schedule (with readings) posted under DOCS Help Room (BPS 1248) is staffed Honors Students received Email with hours January 20, 2006 Physics for Scientists&Engineers 1 2 Coming this week: Kinematics - The description of motion Motion in 1-dimension, a straight line. The concepts of position, displacement, and distance. Computing the position, velocity, acceleration of objects moving in 1-dimension. Mathematical description of motion with constant acceleration. Free fall without air resistance. January 20, 2006 Physics for Scientists&Engineers 1 3 Clicker Survey Question 1: Before this term, what was your most advanced physics course (HS or college)? A. Calculus based B. Non-calculus based C. No physics at all Question 2: What is your class standing? A. B. C. D. Freshman Sophomore Junior Senior or above January 20, 2006 Physics for Scientists&Engineers 1 4 Position 1-dimensional vectors -> one component, x. Use symbol x to denote the position of an object No vector arrows for 1d vectors Position vectors measured relative to the origin (arbitrarily chosen) Vector x can be positive or negative Examples: 7 x = 0.73 m, x = 4' 3 16 ", x = -13.6 km x 0 x X(t) Position vector is a function of time Notation: x(t) Notation: at specific time t1: x(t1)=x1 January 20, 2006 Physics for Scientists&Engineers 1 X(t1) t1 t 5 Graphical Representation Example: Car driving down the road (top view) As car passes the blue dot, we start clock and record its position at regular time intervals Case 1: constant speed Case 2: speeding up Case 3: coming to a halt January 20, 2006 Physics for Scientists&Engineers 1 6 Graphs of the Position Vector Record the location of car's center to obtain a graph of position vector vs. time Typical of motion with constant velocity January 20, 2006 Physics for Scientists&Engineers 1 7 Displacement Displacement = difference between final position and initial position, Dx = x2 - x1 (x1 x(t1 ), x2 x(t 2 )) Displacement is a vector; can be negative Displacement is independent of choice of origin Displacement of going from point b to point a is exactly the negative of going from point a to point b: Dxba = xb - xa = -(xa - xb ) = -Dxab January 20, 2006 Physics for Scientists&Engineers 1 8 Distance The distance = absolute value of displacement l = Dx Distance is always positive (or 0) Distance is a scalar, displacement a vector January 20, 2006 Physics for Scientists&Engineers 1 9 Example: Roundtrip (1) Distance between Des Moines, Iowa, and Iowa City, is listed as 113.5 miles or 182.6 km Straight line, to very good approximation Question: If we take a round trip Des Moines Iowa City Des Moines, what is the total distance and displacement for this trip? January 20, 2006 Physics for Scientists&Engineers 1 10 Example: Roundtrip (2) Answer: Distance: total distance = sum of distance Des Moines - Iowa City plus distance Iowa City - Des Moines. Total distance for round trip = 182.6 km + 182.6 km = 365.2 km. x Displacement: Put origin of coordinate system at Des Moines (does it matter where we put it?) => xD= 0 km Position vector for Iowa City then has value xI=+182.6 km. Displacement vector from Des Moines to Iowa City Displacement vector for return trip DxID = x I - x D = x I - 0 = +182.6 km DxDI = x D - x I = 0 - x I = -182.6 km Total displacement is the sum of both displacements Dxtotal = DxID + DxDI = 182.6 km - 182.6 km = 0 January 20, 2006 Physics for Scientists&Engineers 1 11 Velocity Vector Average velocity Displacement divided by time interval Dx vDt = Dt Units: m/s (Instantaneous) velocity In the limit that time interval approaches 0 Dx dx v(t ) = lim v = lim Dt 0 Dt 0 Dt dt We need derivatives from now on January 20, 2006 Physics for Scientists&Engineers 1 12 Calculus Reminder (from Math Primer) Polynomials: Trig functions: Exponential, log: Product rule: Chain rule: January 20, 2006 d n x = nx n -1 dx d sin ( ax ) = a cos ( ax ) dx d ax e = ae ax dx d 1 ln ( ax ) = dx x d df (x) ^ dg(x) ^ g(x) + f (x) ( f (x)g(x)) = dx ~ dx ~ dx dy dy du y (u ( x)) fi = dx du dx Physics for Scientists&Engineers 1 13 Example: Velocity (1) Between 0 and 10 s, the position vector of a car is given by x(t) = 17.2 m - (10.1 m)(t / s)+(1.1 m)(t /s)2 Question: What is its velocity vector? Answer: Take derivative dx v(t) = dt d = (17.2 m - (10.1 m)(t /s) + (1.1 m)(t /s) 2 ) dt d d d = (17.2 m) - ((10.1 m)(t /s)) + ((1.1 m)(t /s) 2 ) dt dt dt = 0 -10.1 m/s + (2.2 m/s)(t /s) = -10.1 m/s + (2.2 m/s)(t /s) January 20, 2006 Physics for Scientists&Engineers 1 14 Example: Velocity (2) 2 Graph of x(t) = 17.2 m - (10.1 m)(t / s)+(1.1 m)(t /s) and v(t) = -10.1 m/s+(2.2 m/s)(t / s) Note: position at minimum where velocity is zero! Expected from calculus January 20, 2006 Physics for Scientists&Engineers 1 15 Speed Speed is the absolute value of the velocity vector Velocity is a vector, speed a scalar! Relationship to distance: Dx l v = speed = = Dt Dt January 20, 2006 Physics for Scientists&Engineers 1 16 Quiz Question #1 e b c a d At which point(s) does the position equal zero? A) B) C) D) January 20, 2006 Physics for Scientists&Engineers 1 a a b b only and d only & d 17 Quiz question #2 e b c a d A) At which point(s) does the velocity equal zero? B) C) D) E) January 20, 2006 Physics for Scientists&Engineers 1 a b only c only b & d a & d 18 Quiz question #3 e b c a d At which point is the velocity negative? A) B) C) D) E) a b c d e 19 January 20, 2006 Physics for Scientists&Engineers 1 Example: Swimming laps (1) Suppose a swimmer swims the first 50 m of the 100 m freestyle in 38.2 seconds. Once she reaches the far side of the pool, she turns around and swims back to the start in 42.5 seconds. Question: What is the average velocity and speed for the leg from start to the far side of the pool, for the return leg, and for the total trip? January 20, 2006 Physics for Scientists&Engineers 1 20 Example: Swimming laps (2) First Leg: Swimmer starts at x = 0 and swims to x = 50 m. It takes her 38.2 s Average velocity x2 - x1 50 m - 0 m 50 vleg 1 = = = m/s = 1.31 m/s Dt 38.2 s 38.2 Average speed vleg 1 = 1.31 m/s January 20, 2006 Physics for Scientists&Engineers 1 21 Example: Swimming laps (3) Second Leg: Swimmer starts at x = 50 m and swims to x = 0 m. It takes her 42.5 s Average velocity x2 - x1 0 m - 50 m -50 vleg 2 = = = m/s = -1.18 m/s Dt 42.5 s 42.5 Average speed vleg 2 = 1.18 m/s ! Physics for Scientists&Engineers 1 22 January 20, 2006 Example: Swimming laps (4) Entire lap: Swimmer starts at x = 0 m, swims to x = 50 m, and then back to 0. It takes her 38.2 s + 42.5 s = 80.7 s Average velocity: Displacement is 0 => Average velocity = 0 Can also show this by taking the time-weighted average v= vleg 1 Dt1 + vleg 2 Dt2 Dt1 + Dt2 (1.31 m/s)(38.2 s)+( - 1.18 m/s)(42.5 s) = =0 38.2 s+42.5 s Average speed: use total distance = 100 m and total time v = Dx1lap + Dx2lap Dt 100 m = = 1.24 m/s 80.7 s (again same result obtained from weighted average) January 20, 2006 Physics for Scientists&Engineers 1 23 Quiz Question #4 e b c a d At which segment(s) is the acceleration negative? A) B) C) D) a-c c-d c-e d-e January 20, 2006 Physics for Scientists&Engineers 1 24 Quiz Question #5 e b c a d At which point(s) does the acceleration equal zero? A) B) C) D) E) a b c d e 25 January 20, 2006 Physics for Scientists&Engineers 1 ...
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